Tag - Symplectic geometry

Peter Weigel: Positive loops and orderability in contact geometry

4th October, 2013

Orderability of contact manifolds is related in some non-obvious ways to the topology of a contact manifold Σ. We know, for instance, that if Σ admits a 2-subcritical Stein filling, it must be non-orderable. By way of contrast, in this talk I will discuss ways of modifying Liouville structures for high-dimensional Σ so that the result is always orderable. The main technical tool is a Morse-Bott Floer-theoretic growth rate, which has some parallels with Givental's non-linear Maslov index. I will also discuss a generalization to the relative case, and applications to bi-invariant metrics on Cont(Σ).

Dietmar Salamon: GIT and μ-GIT

26th September, 2013

In this lecture I will explain the moment-weight inequality, and its role in the proof of the Hilbert-Mumford numerical criterion for μ-stability. The setting is Hamiltonian group actions on closed Kaehler manifolds. The major ingredients are the moment map μ and the finite-dimensional analogues of the Mabuchi functional and the Futaki invariant.

Richard Kenyon: Dimers and Integrability

29th March, 2013

To any convex integer polygon we associate a Poisson variety, which is essentially the moduli space of connections on line bundles on (certain) bipartite graphs on a torus. There is an underlying integrable Hamiltonian system whose Hamiltonians are weighted sums of dimer covers.

Nancy Hingston: Resonance for Loop Homology of Spheres

15th March, 2013

Fix a metric (Riemannian or Finsler) on a compact manifold M. The critical points of the length function on the free loop space 𝓛M of M are the closed geodesics on M. Filtration by the length function gives a link between the geometry of closed geodesics, and the algebraic structure given by the Chas-Sullivan product on the homology of 𝓛M and the 'dual' loop cohomology product.

If X is a homology class on 𝓛M, the 'minimax' critical level Cr(X) is a critical value of the length function. Gromov proved that if M is simply connected, there are positive constants k and K so that for every homology class X of degree > dim(M) on 𝓛M, k < deg(X)/Cr(X) < K. When M is a sphere, we prove there are positive constants a and b so that for every homology class X on 𝓛M, aCr(X)-b < deg(X) < aCr(X)+b. There are interesting consequences for the length spectrum.

Alvaro Pelayo: Intermediate Symplectic Capacities

1st March, 2013

In 1985 Misha Gromov proved his Nonsqueezing Theorem, and hence constructed the first symplectic 1-capacity. In 1989 Helmut Hofer asked whether symplectic d-capacities exist if 1 < d < n. I will discuss the answer to this question and its relevance in symplectic geometry.